By employing Fourier-Laplace transform

the authors have made a discussion on the multiplying operation between Roumieu ω-test function spaces D_{ω}(R

N

) and Beurling ω-test function spaces D_{ω}

each of them being the sub-spaces of ω-ultra-differentiable spaces;the authors have also presented the relation between the products and convolutions of the elements in the two types of spaces through Fourier transform

proving that D(R

N

) is the multiplicator spaces between D_{ω}(R

N

) and D

ω

.